\(\int x (a+b \arcsin (c x))^{3/2} \, dx\) [179]

Optimal result

Integrand size = 14, antiderivative size = 172 \[ \int x (a+b \arcsin (c x))^{3/2} \, dx=\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{8 c}-\frac {(a+b \arcsin (c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{3/2}-\frac {3 b^{3/2} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 c^2}+\frac {3 b^{3/2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{32 c^2} \]

[Out]

-1/4*(a+b*arcsin(c*x))^(3/2)/c^2+1/2*x^2*(a+b*arcsin(c*x))^(3/2)-3/32*b^(3/2)*cos(2*a/b)*FresnelS(2*(a+b*arcsi
n(c*x))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/c^2+3/32*b^(3/2)*FresnelC(2*(a+b*arcsin(c*x))^(1/2)/b^(1/2)/Pi^(1/2))
*sin(2*a/b)*Pi^(1/2)/c^2+3/8*b*x*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^(1/2)/c

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {4725, 4795, 4737, 4731, 4491, 12, 3387, 3386, 3432, 3385, 3433} \[ \int x (a+b \arcsin (c x))^{3/2} \, dx=\frac {3 \sqrt {\pi } b^{3/2} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 c^2}-\frac {3 \sqrt {\pi } b^{3/2} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 c^2}+\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{8 c}-\frac {(a+b \arcsin (c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{3/2} \]

[In]

Int[x*(a + b*ArcSin[c*x])^(3/2),x]

[Out]

(3*b*x*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(8*c) - (a + b*ArcSin[c*x])^(3/2)/(4*c^2) + (x^2*(a + b*ArcS
in[c*x])^(3/2))/2 - (3*b^(3/2)*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])])
/(32*c^2) + (3*b^(3/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(32*c^2
)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[f*(x^2/d)],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[f*(x^2/d)], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3387

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]

Rule 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 4725

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcSin[c*x])^n/(m
+ 1)), x] - Dist[b*c*(n/(m + 1)), Int[x^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; Fre
eQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 4731

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Sin[-
a/b + x/b]^m*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]

Rule 4795

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 (a+b \arcsin (c x))^{3/2}-\frac {1}{4} (3 b c) \int \frac {x^2 \sqrt {a+b \arcsin (c x)}}{\sqrt {1-c^2 x^2}} \, dx \\ & = \frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{8 c}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{3/2}-\frac {1}{16} \left (3 b^2\right ) \int \frac {x}{\sqrt {a+b \arcsin (c x)}} \, dx-\frac {(3 b) \int \frac {\sqrt {a+b \arcsin (c x)}}{\sqrt {1-c^2 x^2}} \, dx}{8 c} \\ & = \frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{8 c}-\frac {(a+b \arcsin (c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{3/2}+\frac {(3 b) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{16 c^2} \\ & = \frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{8 c}-\frac {(a+b \arcsin (c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{3/2}+\frac {(3 b) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{16 c^2} \\ & = \frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{8 c}-\frac {(a+b \arcsin (c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{3/2}+\frac {(3 b) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{32 c^2} \\ & = \frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{8 c}-\frac {(a+b \arcsin (c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{3/2}-\frac {\left (3 b \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{32 c^2}+\frac {\left (3 b \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{32 c^2} \\ & = \frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{8 c}-\frac {(a+b \arcsin (c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{3/2}-\frac {\left (3 b \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{16 c^2}+\frac {\left (3 b \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{16 c^2} \\ & = \frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{8 c}-\frac {(a+b \arcsin (c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{3/2}-\frac {3 b^{3/2} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 c^2}+\frac {3 b^{3/2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{32 c^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.05 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.73 \[ \int x (a+b \arcsin (c x))^{3/2} \, dx=\frac {b^2 e^{-\frac {2 i a}{b}} \left (\sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {5}{2},-\frac {2 i (a+b \arcsin (c x))}{b}\right )+e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {5}{2},\frac {2 i (a+b \arcsin (c x))}{b}\right )\right )}{16 \sqrt {2} c^2 \sqrt {a+b \arcsin (c x)}} \]

[In]

Integrate[x*(a + b*ArcSin[c*x])^(3/2),x]

[Out]

(b^2*(Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[5/2, ((-2*I)*(a + b*ArcSin[c*x]))/b] + E^(((4*I)*a)/b)*Sqrt[(I*
(a + b*ArcSin[c*x]))/b]*Gamma[5/2, ((2*I)*(a + b*ArcSin[c*x]))/b]))/(16*Sqrt[2]*c^2*E^(((2*I)*a)/b)*Sqrt[a + b
*ArcSin[c*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(280\) vs. \(2(134)=268\).

Time = 0.07 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.63

method	result	size
default	\(-\frac {-3 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, b^{2}-3 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, b^{2}+8 \arcsin \left (c x \right )^{2} \cos \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b^{2}+16 \arcsin \left (c x \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a b +6 \arcsin \left (c x \right ) \sin \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b^{2}+8 \cos \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2}+6 \sin \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a b}{32 c^{2} \sqrt {a +b \arcsin \left (c x \right )}}\)	\(281\)

[In]

int(x*(a+b*arcsin(c*x))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/32/c^2/(a+b*arcsin(c*x))^(1/2)*(-3*(-1/b)^(1/2)*Pi^(1/2)*cos(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2
)*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*b^2-3*(-1/b)^(1/2)*Pi^(1/2)*sin(2*a/b)*FresnelC(2*2^(1/2)
/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*b^2+8*arcsin(c*x)^2*cos(-2*(a+b*arcs
in(c*x))/b+2*a/b)*b^2+16*arcsin(c*x)*cos(-2*(a+b*arcsin(c*x))/b+2*a/b)*a*b+6*arcsin(c*x)*sin(-2*(a+b*arcsin(c*
x))/b+2*a/b)*b^2+8*cos(-2*(a+b*arcsin(c*x))/b+2*a/b)*a^2+6*sin(-2*(a+b*arcsin(c*x))/b+2*a/b)*a*b)

Fricas [F(-2)]

Exception generated. \[ \int x (a+b \arcsin (c x))^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x*(a+b*arcsin(c*x))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int x (a+b \arcsin (c x))^{3/2} \, dx=\int x \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

integrate(x*(a+b*asin(c*x))**(3/2),x)

[Out]

Integral(x*(a + b*asin(c*x))**(3/2), x)

Maxima [F]

\[ \int x (a+b \arcsin (c x))^{3/2} \, dx=\int { {\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}} x \,d x } \]

[In]

integrate(x*(a+b*arcsin(c*x))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*arcsin(c*x) + a)^(3/2)*x, x)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.95 (sec) , antiderivative size = 845, normalized size of antiderivative = 4.91 \[ \int x (a+b \arcsin (c x))^{3/2} \, dx=\text {Too large to display} \]

[In]

integrate(x*(a+b*arcsin(c*x))^(3/2),x, algorithm="giac")

[Out]

1/4*I*sqrt(pi)*a^2*b^(3/2)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) - I*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^
(2*I*a/b)/((b^2 + I*b^3/abs(b))*c^2) - 1/8*sqrt(pi)*a*b^(5/2)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) - I*sqrt(b*
arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b^2 + I*b^3/abs(b))*c^2) - 1/4*I*sqrt(pi)*a^2*b^(3/2)*erf(-sqrt
(b*arcsin(c*x) + a)/sqrt(b) + I*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^2 - I*b^3/abs(b))*c^2
) - 1/8*sqrt(pi)*a*b^(5/2)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) + I*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^
(-2*I*a/b)/((b^2 - I*b^3/abs(b))*c^2) + 1/8*sqrt(pi)*a*b^2*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) - I*sqrt(b*arc
sin(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b^(3/2) + I*b^(5/2)/abs(b))*c^2) + 1/4*I*sqrt(pi)*a^2*b*erf(-sqrt(
b*arcsin(c*x) + a)/sqrt(b) + I*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^(3/2) - I*b^(5/2)/abs(
b))*c^2) + 1/8*sqrt(pi)*a*b^2*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) + I*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))
*e^(-2*I*a/b)/((b^(3/2) - I*b^(5/2)/abs(b))*c^2) - 1/4*I*sqrt(pi)*a^2*sqrt(b)*erf(-sqrt(b*arcsin(c*x) + a)/sqr
t(b) - I*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b + I*b^2/abs(b))*c^2) + 3/64*I*sqrt(pi)*b^(5/2
)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) - I*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b + I*b^2/abs
(b))*c^2) - 3/64*I*sqrt(pi)*b^(5/2)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) + I*sqrt(b*arcsin(c*x) + a)*sqrt(b)/a
bs(b))*e^(-2*I*a/b)/((b - I*b^2/abs(b))*c^2) - 1/8*sqrt(b*arcsin(c*x) + a)*b*arcsin(c*x)*e^(2*I*arcsin(c*x))/c
^2 - 1/8*sqrt(b*arcsin(c*x) + a)*b*arcsin(c*x)*e^(-2*I*arcsin(c*x))/c^2 - 1/8*sqrt(b*arcsin(c*x) + a)*a*e^(2*I
*arcsin(c*x))/c^2 - 3/32*I*sqrt(b*arcsin(c*x) + a)*b*e^(2*I*arcsin(c*x))/c^2 - 1/8*sqrt(b*arcsin(c*x) + a)*a*e
^(-2*I*arcsin(c*x))/c^2 + 3/32*I*sqrt(b*arcsin(c*x) + a)*b*e^(-2*I*arcsin(c*x))/c^2

Mupad [F(-1)]

Timed out. \[ \int x (a+b \arcsin (c x))^{3/2} \, dx=\int x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2} \,d x \]

[In]

int(x*(a + b*asin(c*x))^(3/2),x)

[Out]

int(x*(a + b*asin(c*x))^(3/2), x)